The mechanical structure of a robotic arm system is quite complicated. In analysis of robot kinematics, the mechanical structure is generalized and described as a mechanism parameter set including size (arm length) of mechanical links, connection orientations and angles between joint axes, joint variables, and other geometric variables. The mechanism parameter set is further used to construct a mathematical model for calculating spatial positions of the robotic arm. In other words, according to values of the mechanism parameter set, predictive positioning-points of the robotic arm in space can be calculated by mathematical model.
Hence an ideal mathematical model of the robotic arm can be a function equation F(S) corresponding to a mechanism parameter set S for calculating predictive positioning-points P of the mathematical model of the robotic arm in space. An expression equation is shown below:P≡F(S)
Wherein the mechanism parameter set S includes the size (arm length) of mechanical links, the connection orientations and angles between joint axes, the amount of joint variables, and other geometric variables.
However, in some situations, the actual values of the mechanism parameter set S are different from the ideal values due to influence from factors such as machining tolerances of mechanical components, mechanism assembly error, mechanism transmission error, load stress variation, operation abrasion, and ambient temperature changes. Accordingly, the default values of the mechanism parameter set S tend to have errors such as position deviation ΔP between the actual measured positioning-point N and the predictive positioning-point P of the mathematical model. The position deviation ΔP represents the performance of the robotic arm in terms of positioning accuracy and efficiency, and it also reflects a margin of deviation corresponding to the mechanism parameter set S.
Deviations of every parameter in the mechanism parameter set S are supposed to be a set of mechanism parametric deviations ΔS. The set of mechanism parametric deviations ΔS and the position deviation ΔP are further assumed to have a slight deviation in their linear relationship, as shown below:ΔP=N−P≡J(S)·ΔS 
Wherein the coefficient matrix
      J    ⁡          (      S      )        =            ∂              F        ⁡                  (          S          )                            ∂      S      is a partial differential matrix deriving from the mathematical model F(S) over the mechanism parameter set S.
FIG. 1 is a schematic diagram showing a robotic arm system 10. The robotic arm system 10 comprises a robotic arm 11, a base 12, a storage unit 13, a processing unit 14 and an absolute positioning measuring instrument 15. The robotic arm 11 is disposed on the base 12 and electrically connected to the processing unit 14. The storage unit 13 is used to store a plurality of mechanism parameter sets Sk, k=1, . . . , n (S1˜Sn) and a corresponding plurality of predictive positioning-points Pk, k=1, . . . , n (P1˜Pn). The predictive positioning-point Pk is calculated by substituting the mechanism parameter set Sk into the ideal mathematical model F(S) of the robotic arm 11, and is represented below:Pk≡F(Sk), k=1, . . . ,n 
Wherein the mechanism parameter sets S1˜Sn, are the size (arm length) of mechanical links, the connection orientations and angles between joint axes, the amount of joint variables, and other geometric variables of the robotic arm 11.
The processing unit 14 comprises a calibrating calculation unit 141 and a control unit 142. The processing unit 14 is electrically connected to the storage unit 13. The control unit 142 of the processing unit 14 performs a specific action according to a specific mechanism parameter set S (e.g. Sk), so an end of the robotic arm 11 moves toward a predictive positioning-point P (e.g. Pk) corresponding to the specific mechanism parameter set S.
The absolute positioning measuring instrument 15 can be a coordinate-measuring machine (CMM) or a laser tracker. The absolute positioning measuring instrument 15 is used to perform an absolute positioning measurement on multiple positioning points of the end of the robotic arm 11 such as an end-effector. When the end of the robotic arm 11 moves toward a predictive positioning-point P (e.g. Pk), the absolute positioning measuring instrument 15 obtains a corresponding absolute measured positioning point N (e.g. Nk, k=1, . . . , n).
At this moment, distinct absolute measured positioning points Nk and distinct predictive positioning-points Pk corresponding to n positioning points are repeatedly measured and collected to obtain a linear relationship of the predictive positioning-points Pk and the mechanism parametric deviations ΔS. The linear relationship is shown below:ΔPk=Nk−Pk≡J(Sk)·ΔS, k=1,2, . . . ,n 
According to the above linear relationship derived from enough amounts of positioning points are measured and collected, an optimization equation Φ of the robotic arm 11 is obtained and represented below:
  Φ  =            min              Δ        ⁢                                  ⁢        S              ⁢                  ∑                  k          =          1                n            ⁢                        (                                    Δ              ⁢                                                          ⁢                              P                k                                      -                                                            J                  ⁡                                      (                                          S                      k                                        )                                                  ·                Δ                            ⁢                                                          ⁢              S                                )                2            
Then the processing unit 14 of the robotic arm system 10 utilizes an optimization algorithm and the optimization equation Φ to obtain a set of mechanism parametric deviations ΔS. Finally, the processing unit 14 of the robotic arm system 10 accomplishes calibration by using the set of mechanism parametric deviations ΔS to calibrate the mechanism parameter sets S1˜Sn of the robotic arm 11.
However, the set of mechanism parametric deviations ΔS and the position deviation ΔP are assumed to have a slight deviation in their linear relationship based on approximating the position deviations of a non-linear mathematical model of the robotic arm by a partial differential equation. The approximation method is more effective for small position deviations ΔP. If the position deviations ΔP are too large, the approximation errors would reduce the efficiency of obtaining the set of mechanism parametric deviations ΔS with the optimization equation Φ. In addition, an absolute positioning measuring instrument 15 is required to serve as precision measuring equipment which can perform absolute positioning measurements. An example is the laser tracker. Absolute positioning measuring instruments 15 are expensive and are not easy to be implemented on site in factories.
In view of this, the present application provides a mechanism-parametric-calibration method, wherein calibration measurement embodiments and algorithms are illustrated to obtain a corresponding calculation result for adjusting mechanism parameters of the robotic arm. Accordingly, the accuracy of positioning the robotic arm is improved thereby.